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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Inverse spectral theory for Sturm-Liouville problems with finite spectrum
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by Hans Volkmer and Anton Zettl PDF
Proc. Amer. Math. Soc. 135 (2007), 1129-1132 Request permission

Abstract:

For any positive integer $n$ and any given $n$ distinct real numbers we construct a Sturm-Liouville problem whose spectrum is precisely the given set of $n$ numbers. Such problems are of Atkinson type in the sense that the weight function or the reciprocal of the leading coefficient is identically zero on at least one subinterval.
References
  • F. V. Atkinson, Discrete and continuous boundary problems, Mathematics in Science and Engineering, Vol. 8, Academic Press, New York-London, 1964. MR 0176141
  • P. Binding and H. Volkmer, “Prüfer angle asymptotics for Atkinson’s semi-definite Sturm-Liouville eigenvalue problem”, to appear in Math. Nachr.
  • F. P. Gantmacher and M. G. Krein, Oscillation matrices and kernels and small vibrations of mechanical systems, Revised edition, AMS Chelsea Publishing, Providence, RI, 2002. Translation based on the 1941 Russian original; Edited and with a preface by Alex Eremenko. MR 1908601, DOI 10.1090/chel/345
  • Q. Kong, H. Wu, and A. Zettl, Sturm-Liouville problems with finite spectrum, J. Math. Anal. Appl. 263 (2001), no. 2, 748–762. MR 1866077, DOI 10.1006/jmaa.2001.7661
  • H. Volkmer, “Eigenvalues associated with Borel sets”, to appear in Real Analysis Exchange.
  • Hans Volkmer, Eigenvalue problems of Atkinson, Feller and Krein, and their mutual relationship, Electron. J. Differential Equations (2005), No. 48, 15. MR 2135259
  • Anton Zettl, Sturm-Liouville theory, Mathematical Surveys and Monographs, vol. 121, American Mathematical Society, Providence, RI, 2005. MR 2170950, DOI 10.1090/surv/121
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Additional Information
  • Hans Volkmer
  • Affiliation: Department of Mathematical Sciences, University of Wisconsin, Milwaukee, Wisconsin 53201
  • Email: volkmer@csd.uwm.edu
  • Anton Zettl
  • Affiliation: Department of Mathematics, Northern Illinois University, De Kalb, Illinois 60115
  • Email: zettl@math.niu.edu
  • Received by editor(s): March 21, 2005
  • Received by editor(s) in revised form: November 11, 2005
  • Published electronically: October 11, 2006

    • Dedicated: Dedicated to the memory of F.V. Atkinson
  • Communicated by: Carmen C. Chicone
  • © Copyright 2006 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 135 (2007), 1129-1132
  • MSC (2000): Primary 34B24, 34B09; Secondary 34L05
  • DOI: https://doi.org/10.1090/S0002-9939-06-08563-7
  • MathSciNet review: 2262915